Look familiar? For most people, if they ever learned to do long division properly, they had to “master” something similar to the above procedure/algorithm. What was not required was having the vaguest notion of what’s going on, why anyone should care about knowing how to do it, or why, if you did have some idea about what the answer was supposed to mean, this mumbo-jumbo got you to that answer correctly. Another way of describing the situation is “a black box.” You take two numbers, throw them into the “long division box,” turn the handle as many times as necessary, and – voila! – out pops the magic quotient, with a remainder, if any, in some form or other. Fabulous! Who needs conceptual understanding if you can memorize a procedure and carry it out accurately? Oh, and quickly: don’t forget that “math” (well, school math, anyway), is a race! The “good” students are like machines! The slow, the error-prone, the confused, will be left by the wayside.
Well, that doesn’t work for me anymore with mathematics. I can’t be satisfied with being a flesh-robot or training others to be like that (or abetting teachers to make children view math like that). Procedural fluency sans understanding is not better than using an electronic black box for computation, though traditionalists praise the former and decry the latter. There’s a deep conversation (or pointless, screaming argument complete with epithets that would make participants in a Trump vs. Clinton partisan on-line fight blush) to be had here, but not today. Instead, I’m going to draw back the curtain and show you the wizard hiding behind it.
To begin, let’s recall from last month that one way to think about computing a multiplication problem in the world of integers (Z) is as “repeated addition.” While there are problems with this phrase as a definition of multiplication, one I advise that teachers eschew, there is no problem getting the product of, say, 18 x 6 by summing six addends of eighteen or vice versa. Not ideal for efficiency, of course.
By analogy, since division is the inverse operation of multiplication and subtraction is the inverse operation of addition, we should be able to compute division in Z as repeated subtraction (as long as we are comfortable with remainders. In the rational numbers (Q), we have “fractions” or decimals when a division of integers doesn’t give a “nice” integer quotient). Here’s an example of how that would work in practice:
So, we could certainly do without long division if we’re willing to do repeated subtraction as many times as necessary to get the quotient of two integers, right? And if we are not in a hurry for results or worried that adding more than the minimum calculations necessary increases the likelihood of error, we’re fine. We really DO have that option.
But for those who prefer another approach, there’s long division. Why does it work? That’s the missing piece for most of us, including, I strongly suspect, most K-6 math teachers. I would be willing to bet that if you asked people to explain why the long-division algorithm works – not simply what the steps are, but why and how they make sense – few could give a coherent, correct response.
So, let’s try to unpack long division by applying the standard algorithm to something similar to the previous problem (we can risk a bigger dividend here):
I trust your answer agrees with mine. But how do we know we are correct? Well, 78 x 22 = 1716. Add the remainder, 14, and we get 1730, the original dividend. Math works, life is good.
Except that we don’t know why we got that result, exactly, only that we followed the recipe and almost miraculously, out popped the perfect cake.
One thing worth noting that could easily be missed: if we add 1540 + 176, we get 1716. That can’t be sheer coincidence, can it? What are those numbers, anyway?
Let’s go through the steps used to get our answer, but a bit more thoroughly than you are probably used to doing. First, you likely said, “Hmm, 22 won’t go into 17, so let’s try 22 into 173. Ah, that goes, um, 8 times. Wait, no, too big, so try 7; yeah, cool, that’s 154.”
All well and good, but mostly lies. First, that’s not “17,” it’s 1700. The question is NOT really “What is 22 into 17?” but rather, “how many groups of 1000 x 22 are there in 1730?” Oh, that surprises you? Well, why do you take for granted that you don’t have to ask that? The fact is that only experience, number sense or the mindless following of what teachers, parents, or peers told you lets you skip that question as unneeded. And similarly, probably without knowing it, you miss that the first question about 22 into 17 now becomes “How many groups of 100 x 22 are there in 1730?”
Again, the answer is 0, since 1 x 100 x 22 = 2200 which is too big. So finally, we should ask, “How many groups of 10 x 22 are there in 1730?” And the answer is 7. But please note well: that 7 represents “70,” which is 7 x 10, which is what we really multiply 22 by to get the first “partial product” of 1540 (see that I appended a zero to the 154 you calculated to make clear what is really being subtracted from 1730 on the first pass). Also, realize that using our method, that is the maximum we can justify subtracting, but we might have subtracted less! (See the discussion of partial quotients below).
So after we subtract 1540, we have not 19, but 190 remaining. The rest is pretty obvious (a dangerous word I’ll risk using here). The next step does not require lying: we really do ask,”How many groups of 1 x 22 are there in 190?” The answer is indeed 8. Any more would put us past 190. Our second partial product, from 8 x 22, is 176. We subtract that from 190 and are left with 14, too small for any more groups of 22 to be removed. That’s our remainder, then (alternatively, we can say that there are 14/22 of a group left, where a whole group has 22 in it).
So that is almost all we need to note save that what is missing from the process you were taught in school takes for granted one crucial idea: place value. We are first taking 70 groups of 22 from 1730 all at once, rather than doing it 70 times via individual subtractions of 22 at a time. And that’s a very good thing indeed. But because this process of compressing 70 subtractions into one estimation, one multiplication (or more if we estimate badly) and one subtraction hides information from us. “Better” algorithms are designed for efficiency, not for transparency. You snooze, you lose.
Except, we pretty much all snooze because it is almost unheard of for teachers to take the time to unpack this process, let alone teach it as repeated subtraction, then as partial quotients using place value, and then finally going to the more efficient but opaque standard algorithm. So for kids who don’t pick up the procedure, practice it to “mastery,” and then move on to the next disconnected procedure, long division is frequently a stumbling block. Even some students who can perform the algorithm correctly find it off-putting, I believe, given how negatively they react to the introduction of polynomial long division, a procedure that is, in practice, far easier to perform.
I would be remiss to conclude this essay without looking at one other approach, partial quotients: a blend of repeated subtraction and the standard long division algorithm. Let’s take the second example again, 1730 ÷ 22.
What if we weren’t sure about the exact procedure for the standard algorithm, but we felt that we knew the point of division (which is actually a lot more complicated and context-dependent than I have chosen to explore in this post): how many groups of 22 are there in 1730 with a leftover amount of from 0 to fewer than 22? (In general, we could call 22 the divisor, 1730 the dividend, 78 the quotient, and 14 the remainder, and in symbols, b = a*q + r, where b is the dividend, a is the divisor, q is the quotient, and r is the remainder, a, b, q, and r in Z, and 0 ≤ q < a).
So perhaps we do this:
What is THAT all about? Well, I thought, “Hmm, 10 x 22 = 220, so 20 x 22 = 440.” I then subtracted 1 x 20 x 22 and was able to repeat that twice more. But that left me with 410, too small for another subtraction of 20 x 22, but big enough to subtract 1 x 10 x 22. That left 190, too small for subtracting another 10 x 20 (predictably, given the previous analysis), but big enough to take away 1 x 5 x 22, or 110. That left 80, and I could see that 3 x 22 = 66 was small enough, while 4 x 22 = 88 would be too much to subtract. And with 14 left, I had to stop subtracting. Adding the partial quotients gave the complete quotient of 78 (groups of 22) plus the remainder, 14.
Before the forces of righteousness attack me for “dumbing down” America’s youth, I am not lobbying for “partial quotients” to replace the standard algorithm. Nor am I arguing against having students learn the standard algorithm, since, after all, calculators, computers, and smartphones do arithmetic, and much more, faster and more accurately. Instead, I am advocating for the end to black boxes when understanding is readily provided. How we get there, however, I will leave for another column.
Michael. how about the ‘rule of false position’ or guess and modify?
With 173 divided by 22, we guess a quotient of 10. Yet when we check, we find 22 divisor × 10 quotient = 220 dividend. Darn! We’re over the original dividend by 47. No big deal. Just take away two of the 22s and we get a quotient of 8, yet our dividend is still 3 too much. OK, we’ll have to make the quotient 7 with 19 remaining.
We have two answers, 7 + 19/22 and 8 – 3/22. Convention says we ignore the latter.
You might find it interesting to look at how the rest of the world does long division. If I remember correctly at least 20 years ago most of the world considered our long division algorithm more than strange.
In any case I used to momentarily infuriate (smile) my math education students by telling them that every step in some instance of the standard long division algorithm was a mathematical lie and that I had a nice bridge for sale. The decimal version is even worse and, a quick glace at US websites purporting to teach such, indicate some lingering problems.
I’m not sure whether they will find it to be of interest but I thought that some of your viewers might like to see how my “adjective/noun” theme treats multiplication and division. For that reason I have included the links for the two lectures I use in my workshops for elementary school teachers. The website address is http://www.mathasasecondlanguage.org and the two vudeos I mentioned are at https://www.youtube.com/watch?v=b8Zjth1ZVy8&index=12&list=PLWwhzNbWxQjvu7zDZhLytz9iUXI8IUN3D
I teach third grade. Using color tiles, we build arrays. If we build a 3×5, it is very easy for students to see that 3×5=15. To transfer that idea to division, we take 15 color tiles and arrange them into 3 equal rows. The array now has an unknown quotient, which they can determine using the manipulatives. Then we get 16 color tiles. Can they be arranged into 3 equal rows? Then we delve into the meaning of the 16th tile. What would we do with it? We do this over and over, with different numbers, hoping that the meaning eventually becomes strong enough for more abstract applications.
Hi Jacquelyn. I believe that what you have written is important.. The use of ties is a great aid for helping students internalize mathematical concepts. For example, people will look at 2 X 7 = 7 X 2 and say such things as “It’s obvious because all we did was change the order”, etc. However when we write out what the equality says in expanded form we see the not so obvious equality:
2 + 2 + 2 + 2 + 2 + 2 + 2 = 7 + 7
However if we arrange the tiles in a rectangular array consisting of 2 rows each with 7 tiles, we see almost instantly that there are also 7 columns each with 2 tiles. With respect to your illustration, 5 rows each with 3 tiles is the same number of tiles as 3 columns each with 5 tiles.
More generally, if we use tiles, we can replace the axioms of whole number arithmetic by one simple axioms. Namely, the number of tiles in group does not depend on the order in which they are counted nor in the way that they are arranged.
Based on my own anecdotal experiences this format works very well and I use it in all of my workshops with elementary school teachers. In fact, you can see the video version of what I cover in my live workshops by clicking on
to see the playlist.
In closing I send you my best wishes and hope that you are experiencing great satisfaction in your role of an elementary school teacher.
Nice videos Herb. It is no coincidence that all the underlined word’s in Michael’s ‘The Steps to Long Division’ graphic are VERBS. The four basic operations are also verbs. Herb, to focus on adjectives and nouns may be to completely misrepresent the foundations of elementary (intuitive) mathematical logic developed by the ancient Chinese and Indians.
The first mathasasecondlanguage slideshow I looked asks ‘what must you do’ questions, yet gives nouns as answers instead of verbs. A multiple choice example follows.
Q. “If you move 32 units to the right of where you were, what must you do to return to your starting point?”
A. [⁻32 units] [⁺32 units]
Source: the Directed Distance Model Slide 25.
The book ‘The Problem with Math is English’ by Concepcion Molina also discusses the parts of speech as they can be applied to basic math, yet also fails to correctly explain the VERBS in integer arithmetic. Too much math ed is static when it needs to be dynamic. Verb rhymes with Herb! Only by clearly explaining the verb/actions of the ancient Chinese and Indians, as applied to quantities, will kids ever come to an intuitive understanding of basic mathematics.
As for the unfortunate MIRA* video promoted above, titled ‘Multiplying (VERB) Whole (ADJECTIVE) Numbers (NOUNS), it is clear VERBS take precedence over adjectives and nouns!
Slide 23 on in the slideshow available via https://www.jonathancrabtree.com/LLEM/ explains the verbs, adjectives, and nouns involved with integer multiplication correctly as the ancient Chinese and Indians intended. I hope you take a look as I’m sure you will be able to find areas of improvement in my work as well.
Continuing the departure from long division, the secret math hack the Chinese and Indians leveraged was to not use the adjectives ‘positive’ and ‘negative’! Consistent with the concept of Yang and Yin, with two opposing units ever-present, basic math becomes child’s play! The direct link to the slideshow explaining integer multiplication is https://1drv.ms/p/s!AiiJ6XgphELidETf6CoiWWpuGec The peer-reviewed conference paper associated with the slideshow is at https://www.bit.ly/LostLogicOfMath
Again, I am an advocate of clarifying basic math by making the parts of speech explicit. Yet current-day advocates are hardly the first. The first English language mathematics books directly connected the vocabulary of the new-fangled base ten math with zero (replacing Roman Numerals) with the parts of speech. If you are interested, I’d be happy to share my research on the earliest English use of the ‘parts of speech’ in math education with you, Herb. Let me know.
mathpedagogy-research (at) yahoo (dot) com
* Multiplication Is Repeated Addition
Actually my use of “adjective/noun” might be a bit deceiving. It would have been more accurate if I had said that students visualize numbers in terms of quantities. A quantity is a noun phrase that consists of an adjective (usually what kids call numbers) and a noun (usually referred to as a unit). In essence, we have seen quantities such as 3 people, 3 apples, 3 tally marks but never “threeness” by itself. If you look at some of my other videos (the playlist is at https://www.youtube.com/watch?v=8TaLSX7xybA&list=PLufObkSlzUUU4oKivkiwchXBRfAzQXpcu ) you will see how viewing arithmetic in terms of quantities rather than numbers makes it much easier for students to internalize the more abstract arithmetic concepts.
I would love to see some of your own work but as a caveat I should point out that I am working diligently to complete my website and at age 88 it is difficult for me to multi-task. So it is possible that I might note give your work the amount of attention it deserves.
Herb & Jonathan,
This has been a rather mad week or so for me. Aside from needing to complete and post my three April pieces, the last one of which went up around 2:30 AM Sunday morning (EST), I had an on-line calculus final to take (long story) under aggravating conditions and for which I did very little studying, and a lot of loose ends to wrap up before my son and I drive to NJ on Thursday for a Saturday celebration of my mother’s 90th birthday (which was in early March; my brothers are based in the SF Bay area and didn’t want to risk traveling east in the winter), as well as my aunt’s impending 97th birthday later this month. I mention their nonagenarian anniversaries because I’m counting on Herb to be with us for a while to come so that he can continue to make his unparalleled contributions to mathematics education. Maybe I’ll be able to conduct an anniversary interview with him in, say, five years.
One thing that’s intriguing to me in the wake of responses to my division blog post (somehow, the one on long multiplication didn’t engender much reaction) is how there really thoughtful folks working around the world to make more sense of basic math (for those who know the 3Blue1Brown YouTube channel, its creator has just launched a ten-day series on “The Essence of Calculus” that I already have fallen behind on; Jonathan might recognize the name Kirby Urner, who is doing intriguing work with “Martian Mathematics” in the Portland, OR area, as well as Norman Wildberger, a near-mirror of James Tanton in being a Canadian teaching mathematics in Australia and with his own fascinating YouTube channel chock full of very different takes on a wide range of mathematical topics. I could go on, but I think the point is made: we’re not alone. And if this blog accomplishes nothing else in the years to come, I hope that the interviews I’m doing will help people see that there’s an amazing array of mathematicians, mathematics education researchers, teacher educators, K-12 math teachers, and bloggers who are not constrained by what I consider to be forces of evil (yes, I really mean that, though not in any religious sense) who seek to ensure that nothing changes in how we present mathematics to learners of all ages. Without wanting to crack open the Math Wars here quite yet, I keep bumping into some truly reactionary thinkers who seem to think that there’s precisely one way to think about and teach math, and it isn’t what any of us are doing because that’s not the way they learned 20, 30, 40, or more years ago.
I’ll close this comment by saying that I received an email from a friend a couple of hours ago, someone I met my freshman year in college in 1968 who gave me the greatest compliment of my professional life in response to the piece I posted on long-division. I’m hoping he will take my request to add it as a comment here or give me permission to post it myself. I’ll simply say now that when a 68-year-old reads something I wrote and responds as he did, I know I made the right decision to leave the world of literature and instead pursue mathematics education in my mid-30s. Not that I didn’t love literature (and still do), but rather that I’m in a world that needs me and you and others I’ve mentioned and many more I haven’t (some of whom, I’m thrilled to note, I’ll be interviewing between now and next January), who are committed to wresting the reins of mathematics education away from people who are unwilling and/or unable to let go of their egos and arrogance sufficiently to better serve the needs of the larger community of “students” (of which I’m a proud, if not always diligent member) when it comes to mathematics. Let’s agree to work together to make the table of mathematics as welcoming as possible for everyone who, like my college friend, has long believed they could never sit.
There’s a great piece in the Homeschooler section of my book, Playing with Math, where a teen has a problem she wants to figure out, and she tries to do it by repeated subtraction because there was no need in her life for long division before this and she doesn’t know it. When her “subtraction towers” are giving her grief, mom (a college math teacher) asks if she’s like to learn long division. (Even though her explanation of the procedure doesn’t go into meaning, her simpler examples link the procedure to its meaning.)
The piece is called Radically Sensible Ideas. You can look at an online copy of the book for free (and you might love it enough to want a hard copy, $19). https://naturalmath.com/playingwithmath/
Michael Paul Goldenberg, I agree with the sentiment of this article and, in fact, usually teach using all the aforementioned methods. However, since not all my students master division, at least in the short term that I work with them, let me play devil’s advocate or at least ponder that understanding division processes aren’t greatly important; at least at first.
I call this the “Wax on. Wax off.” theory, in reference to the Karate Kid. Mr. Miyagi tells his prodigee to not ask questions, just do as told. The idea is that once the kid has the movements ingrained in muscle memory, he will understand the “why” of what he has learned later. So, with some students who struggle with number sense, perhaps I should just tell them to memorize the steps in the algorithm and not even consider why they do it this way. After they can perform the process, could I not then explain better what it all means?
I received the following by email from Gerry Fischer, my closest friend in college. It is the greatest professional compliment I’ve ever gotten, and it touches upon precisely why I decided to pursue mathematics education, as well as why I am excited about working on this blog:
Hope you are well this evening.
So, I read this article.
I am barely literate in basic math.
Even thinking about it is something I learned to avoid many decades ago.
Your article was great!
It was interesting.
It was understandable.
It leads me to believe – perhaps mistakenly – that I could actually do this….
And know what I was doing.
I’m sixty-eight years old and I Don’t get to feel that way about very much, very often.
So thanks for writing it – and sending it.”
I always teach my students that there are only two mathematical operations
Addition and subtraction
Multiplication is jus a short-cut form for addition and division is a shortcut for subtraction.
I totally agree with your post
In a way you could make case hat there is only addition; at least in the sense that 5 – 3 = ___ is another form of saying that 3 + __ = 5.
However, in dealing with fractions, when it comes to multiplication and division we have to be a bit more careful. For example I don’t believe there is a solid case for saying that multiplication is repeated addition when we are computing, say, 12 X 13.
I’ll be interviewing Keith Devlin this month and plan to raise the “Multiplication IS Repeated Addition” (MIRA) issue with him. Neither of us believes that’s an accurate representation of the essence of multiplication. I’d be more inclined to say that there are two basic operations in arithmetic: addition and multiplication. I don’t want to get too deeply into all of that here (though others should feel free to do so!), but I’ll just ask this for now: why must we find common denominators to do addition or its inverse in the rational numbers, but not do multiplication or its inverse?
I have at least two more related examples to chew on that I will raise in my interview and/or in this month’s piece about arithmetic.
Well I guess you could have a lot of fun with, say, π X e or some such thing.
As I see it, using tally marks is fine hen your restricting your attention to whole number arithmetic. However once you want to generalize it beyond that you have to (or at least, should) replace the tally marks by rectangles and introduce the area model.
Hello! I’m glad to find this article. I think that when dealing with things that are abstract, it’s very important to examine the paradigms and mindsets used to approach the abstraction. As such, it’s always refreshing to read a different perspective than the norm.
Your comment about polynomial long division reminded me of when I first encountered the tool, in high school Pre-calculus. At that point, I and most of my classmates had entirely forgotten what long division was and the whole thing seemed very unintuitive compared to the rest of the course, which was generally built on concepts from the past few years. Long division had been left behind years earlier in elementary school!
The division of polynomials becomes much easier to understand if we use place value notation in which the powers of x replace the powers of 10. For example 534 would represent 5x^2 + 3x + 4. The only difference is that the coefficients need not be the whole numbers 0 through 9; and that subtraction has to be done by the “add the opposite” rule. You may enjoy looking at my website which can be viewed by clicking on the link https://www.adjectivenounmath.com/.
And if you click on https://archive.org/details/ClassicAlgebraDivisionOfPolynomials you will access my lecture on polynomial division. The major possible drawback to doing this is that the lectures are sequential and looking specifically at one lecture without having seen the previous ones might make things look confusing to you (but I doubt it in this case).
I’m not sure whether my cpmment is useful to you but I felt like writing it anyway to share it with others.
I think it is really interesting to repeatedly subtract when dividing just like how we repeatedly add when we multiply. I haven’t really thought of that, I just do it and really not think about the process anymore, thanks for that little trick, can be useful to show someone how to divide.
I think it is very interesting to think of long division as repeated subtraction, like how multiplication is repeated addition. I think this can be really useful when it comes to showing someone how to divide. I usually just go through and divide without thinking much about it, thanks for the blog.
I think that a lot of us don’t fully comprehend the inner working of long division because there’s generally a lack of mention of the placeholders along with the repeated subtraction aspect. So when it gets to polynomial division, it can become more nebulous as a result.
For a rigorous/ higher-math style to integer division, this guide might be of help. It goes over the theory behind a typical Euclidean division procedure algebraically, and introduces some of the better or lesser known alternatives to long division as well.